cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A339382 Number of partitions of n into an even number of distinct primes (counting 1 as a prime).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 6, 5, 5, 5, 6, 6, 7, 7, 9, 8, 9, 8, 11, 10, 11, 12, 14, 12, 15, 14, 17, 16, 17, 17, 22, 20, 22, 21, 25, 24, 28, 27, 31, 30, 33, 31, 39, 36, 40, 40, 46, 42, 49, 47, 54, 53, 58, 55, 67, 63, 70, 68
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].

Links

Crossrefs

Cf. A000586, A008578, A036497, A067661, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339383.

Programs

  • Maple
    s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
      `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
       b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 02 2020
  • Mathematica
    nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] + (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * ((1 + x) * Product_{k>=1} (1 + x^prime(k)) + (1 - x) * Product_{k>=1} (1 - x^prime(k))).
a(n) = (A036497(n) + A298602(n)) / 2.

A339383 Number of partitions of n into an odd number of distinct primes (counting 1 as a prime).

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 6, 6, 5, 7, 6, 8, 7, 8, 9, 10, 9, 12, 11, 12, 11, 14, 14, 16, 15, 17, 17, 20, 17, 21, 22, 24, 22, 27, 25, 30, 28, 31, 31, 36, 33, 40, 39, 42, 40, 47, 46, 53, 49, 55, 54, 63, 58, 68, 67, 73
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(21) = 4 because we have [17, 3, 1], [13, 7, 1], [13, 5, 3] and [11, 7, 3].

Links

Crossrefs

Cf. A000586, A008578, A036497, A067659, A184171, A184172, A184198, A184199, A298602, A339380, A339381, A339382.

Programs

  • Maple
    s:= proc(n) option remember;
      `if`(n<1, n+1, ithprime(n)+s(n-1))
    end:
b:= proc(n, i, t) option remember; (p-> `if`(n=0, t,
      `if`(n>s(i), 0, b(n, i-1, t)+ `if`(p>n, 0,
       b(n-p, i-1, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 02 2020
  • Mathematica
    nmax = 75; CoefficientList[Series[(1/2) ((1 + x) Product[(1 + x^Prime[k]), {k, 1, nmax}] - (1 - x) Product[(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * ((1 + x) * Product_{k>=1} (1 + x^prime(k)) - (1 - x) * Product_{k>=1} (1 - x^prime(k))).
a(n) = (A036497(n) - A298602(n)) / 2.

A338826 G.f.: (1/(1 + x)) * Product_{k>=1} 1/(1 + x^prime(k)).

Original entry on oeis.org

1, -1, 0, -1, 2, -2, 2, -3, 4, -4, 5, -7, 8, -9, 11, -13, 15, -18, 21, -24, 28, -32, 37, -43, 49, -55, 63, -72, 81, -92, 104, -117, 131, -147, 166, -185, 206, -231, 257, -285, 317, -353, 391, -432, 478, -528, 583, -643, 708, -778, 855, -940, 1031, -1130, 1238, -1354
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Comments

The difference between the number of partitions of n into an even number of prime parts (including 1) and the number of partitions of n into an odd number of prime parts (including 1).
Convolution inverse of A036497.

Crossrefs

Cf. A000586, A000607, A008578, A036497, A046675, A048165, A298602.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[(1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[k, (-1)^(k/#) # &, PrimeQ[#] || # == 1 &] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 55}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * A048165(k).
Showing 1-3 of 3 results.

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